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Properties

Label 16.0.16939126620...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 5^{12}\cdot 31^{2}\cdot 41^{2}$
Root discriminant $32.68$
Ramified primes $2, 5, 31, 41$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T797

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2911, 7184, 26872, 22072, 15492, 2564, -4088, -4224, -1360, 144, 514, 192, -3, -32, -8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 - 32*x^13 - 3*x^12 + 192*x^11 + 514*x^10 + 144*x^9 - 1360*x^8 - 4224*x^7 - 4088*x^6 + 2564*x^5 + 15492*x^4 + 22072*x^3 + 26872*x^2 + 7184*x + 2911)
 
gp: K = bnfinit(x^16 - 8*x^14 - 32*x^13 - 3*x^12 + 192*x^11 + 514*x^10 + 144*x^9 - 1360*x^8 - 4224*x^7 - 4088*x^6 + 2564*x^5 + 15492*x^4 + 22072*x^3 + 26872*x^2 + 7184*x + 2911, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{14} - 32 x^{13} - 3 x^{12} + 192 x^{11} + 514 x^{10} + 144 x^{9} - 1360 x^{8} - 4224 x^{7} - 4088 x^{6} + 2564 x^{5} + 15492 x^{4} + 22072 x^{3} + 26872 x^{2} + 7184 x + 2911 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1693912662016000000000000=2^{32}\cdot 5^{12}\cdot 31^{2}\cdot 41^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.68$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 5, 31, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{1524892163270495571864480590371} a^{15} - \frac{709592134689345915662308970168}{1524892163270495571864480590371} a^{14} + \frac{454434478493254319350874151334}{1524892163270495571864480590371} a^{13} + \frac{260385480491331263122777862206}{1524892163270495571864480590371} a^{12} + \frac{169663731085623886652026618510}{1524892163270495571864480590371} a^{11} - \frac{184024725370473669911434971075}{1524892163270495571864480590371} a^{10} + \frac{346539338549875286425240458711}{1524892163270495571864480590371} a^{9} + \frac{120072376873219475610385667704}{1524892163270495571864480590371} a^{8} + \frac{349595384404083809888238160434}{1524892163270495571864480590371} a^{7} + \frac{275738154748818617102131532595}{1524892163270495571864480590371} a^{6} + \frac{677981519519411409798749672320}{1524892163270495571864480590371} a^{5} + \frac{641941589348866567445045510753}{1524892163270495571864480590371} a^{4} - \frac{609278190917838207697904348090}{1524892163270495571864480590371} a^{3} + \frac{1077270705755172535814387935}{37192491787085257850353185131} a^{2} + \frac{102874011556446094458886759061}{1524892163270495571864480590371} a - \frac{6636473573838622070168668069}{37192491787085257850353185131}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{120399321013845297938624}{686579091972307776616155151} a^{15} + \frac{83403199279560875694912}{686579091972307776616155151} a^{14} + \frac{1001144896718085868756196}{686579091972307776616155151} a^{13} + \frac{3073010468405636390123851}{686579091972307776616155151} a^{12} - \frac{2500005366398853417717080}{686579091972307776616155151} a^{11} - \frac{23887922607190298135675218}{686579091972307776616155151} a^{10} - \frac{41592934236041683236159244}{686579091972307776616155151} a^{9} + \frac{29899460613630500209645149}{686579091972307776616155151} a^{8} + \frac{170947167150850891084165072}{686579091972307776616155151} a^{7} + \frac{332342605963645827777097732}{686579091972307776616155151} a^{6} + \frac{104660820285829389238680028}{686579091972307776616155151} a^{5} - \frac{544710511467048437208057001}{686579091972307776616155151} a^{4} - \frac{1297488186240623438321116120}{686579091972307776616155151} a^{3} - \frac{29178917492664806038453718}{16745831511519701868686711} a^{2} - \frac{1482066513667932346650106040}{686579091972307776616155151} a + \frac{7548475222369627289401929}{16745831511519701868686711} \) (order $10$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 698642.993125 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T797:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 62 conjugacy class representatives for t16n797 are not computed
Character table for t16n797 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 4.0.8000.2, \(\Q(\zeta_{5})\), \(\Q(\sqrt{2}, \sqrt{5})\), 8.0.64000000.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
31Data not computed
41Data not computed